A stream of water strikes a stationary turbine blade horizontally, as the drawing illustrates. The incident water stream has a velocity of + 17.4 m/s, while the exiting water stream has a velocity of – 15.0 m/s. The mass of water per second that strikes the blade is 33.6 kg/s. Find the magnitude of the average force exerted on the water by the blade.
To find the magnitude of the average force exerted on the water by the blade, we can use the conservation of linear momentum.
The formula for linear momentum is given by:
Momentum = mass × velocity
Initially, the momentum of the incoming water stream is given by:
Initial Momentum = mass × initial velocity
Similarly, the momentum of the exiting water stream is given by:
Final Momentum = mass × final velocity
According to the conservation of linear momentum:
Initial Momentum = Final Momentum
Therefore, we can write the following equation:
mass × initial velocity = mass × final velocity
Simplifying the equation, we get:
mass × (initial velocity - final velocity) = 0
Now, we can substitute the given values into the equation:
33.6 kg/s × (17.4 m/s - (-15.0 m/s)) = 0
Calculating this equation, we have:
33.6 kg/s × (32.4 m/s) = 1088.64 kg·m/s
This value represents the change in momentum per second of the water stream.
Since force is defined as the rate of change of momentum, we can calculate the force exerted on the water by dividing the change in momentum by the time interval taken to produce that change.
However, the time interval is not given in the question. Therefore, we cannot determine the magnitude of the average force exerted on the water by the blade without the time interval.
To find the magnitude of the average force exerted on the water by the blade, we can use the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum before an event is equal to the total momentum after the event, assuming there are no external forces acting on the system.
Let's denote the mass of water striking the blade per second as m, the velocity of the incident water stream as v1, and the velocity of the exiting water stream as v2.
According to the principle of conservation of momentum, the total momentum before the event is equal to the total momentum after the event:
m * v1 = m * v2 + F * Δt
where F is the force exerted by the blade on the water, and Δt is the time interval during which the force is applied.
From the given information, we have:
m = 33.6 kg/s (mass of water striking the blade per second)
v1 = 17.4 m/s
v2 = -15.0 m/s (negative sign indicates opposite direction of motion)
The sign convention for velocities is important in this case. Since the incident water stream is moving in the positive direction, its velocity is positive. However, the exiting water stream is moving in the opposite direction, so its velocity is negative.
Solving the equation for the force F, we have:
F = (m * (v1 - v2)) / Δt
To calculate the magnitude of the average force, we need to find the value of (v1 - v2) and the time interval Δt. Unfortunately, these values are not provided in the given information. We would require additional information to calculate the average force exerted on the water by the blade.